MACS 30200
University of Chicago
\[Y = \beta_0 + \beta_1 X + \beta_2 Z + e_i\]
\[E(Y) = \beta_0 + \beta_1 X + \beta_2 Z\]
\[\frac{\delta E(Y)}{\delta X} = \beta_1\]
\[\frac{\delta E(Y)}{\delta Z} = \beta_2\]
\[Y = \beta_0 + \beta_1 X + \beta_2 Z + \beta_3 XZ + e_i\]
\[ \begin{split} E(Y) & = \beta_0 + \beta_1 X + \beta_2 Z + \beta_3 XZ \\ & = \beta_0 + \beta_2 Z + (\beta_1 + \beta_3 Z) X \end{split} \]
\[\frac{\delta E(Y)}{\delta X} = \beta_1 + \beta_3 Z\]
\[E(Y) = \beta_0 + \beta_2 Z + \psi_1 X\]
\[ \begin{split} E(Y) & = \beta_0 + \beta_1 X + (\beta_2 + \beta_3 X) Z \\ & = \beta_0 + \beta_1 X + \psi_2 Z \end{split} \]
If \(Z = 0\), then:
\[ \begin{split} E(Y) & = \beta_0 + \beta_1 X + \beta_2 (0) + \beta_3 X (0) \\ & = \beta_0 + \beta_1 X \end{split} \]
If \(X = 0\), then:
\[ \begin{split} E(Y) & = \beta_0 + \beta_1 (0) + \beta_2 Z + \beta_3 (0) Z \\ & = \beta_0 + \beta_2 Z \end{split} \]\(\psi_1\) and \(\psi_2\)
Obtaining estimates of parameters
\[\hat{\psi}_1 = \hat{\beta}_1 + \hat{\beta}_3 Z\] \[\hat{\psi}_2 = \hat{\beta}_2 + \hat{\beta}_3 X\]Obtaining estimates of standard errors
\[\widehat{\text{Var}(\hat{\psi}_1}) = \widehat{\text{Var} (\hat{\beta}_1)} +Z^2 \widehat{\text{Var} (\hat{\beta}_3)} + 2 Z \widehat{\text{Cov} (\hat{\beta}_1, \hat{\beta}_3)}\]
\[\widehat{\text{Var}(\hat{\psi}_2}) = \widehat{\text{Var} (\hat{\beta}_2)} + X^2 \widehat{\text{Var} (\hat{\beta}_3)} + 2 X \widehat{\text{Cov} (\hat{\beta}_2, \hat{\beta}_3)}\]
\[Y = \beta_0 + \beta_1 D_1 + \beta_2 D_2 + \beta_3 D_1 D_2 + e_i\]
\[ \begin{split} E(Y | D_1 = 0, D_2 = 0) & = \beta_0 \\ E(Y | D_1 = 1, D_2 = 0) & = \beta_0 + \beta_1 \\ E(Y | D_1 = 0, D_2 = 1) & = \beta_0 + \beta_2 \\ E(Y | D_1 = 1, D_2 = 1) & = \beta_0 + \beta_1 + \beta_2 + \beta_3 \\ \end{split} \]
\[Y = \beta_0 + \beta_1 X + \beta_2 D + \beta_3 XD + e_i\]
\[ \begin{split} E(Y | X, D = 0) & = \beta_0 + \beta_1 X \\ E(Y | X, D = 1) & = (\beta_0 + \beta_2) + (\beta_1 + \beta_3) X \end{split} \]
\[Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2 + e_i\]
\[Y = \beta_0 + \beta_1 X + \beta_2 X^2 + e\]
\[\frac{\delta E(Y)}{\delta X} = \beta_1 + 2 \beta_2 X\]
\[ \begin{align} Y = \beta_0 &+ \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_4 X_1 X_2 \\ & + \beta_5 X_1 X_3 + \beta_6 X_2 X_3 + \beta_7 X_1 X_2 X_3 + e \end{align} \]
\[ \begin{align} Y = \beta_0 &+ \beta_1 X + \beta_2 D_1 + \beta_3 D_2 + \beta_4 X D_1 \\ & + \beta_5 X D_2 + \beta_6 D_1 D_2 + \beta_7 X D_1 D_2 + e \end{align} \]
ObamaTherm)RConservObamaConservGOP## ObamaTherm RConserv ObamaConserv GOP
## Min. : 0.0 Min. :1.00 Min. :1.00 Min. :0.00
## 1st Qu.: 50.0 1st Qu.:2.00 1st Qu.:2.00 1st Qu.:0.00
## Median : 75.0 Median :5.00 Median :2.00 Median :0.00
## Mean : 69.6 Mean :4.24 Mean :2.98 Mean :0.24
## 3rd Qu.:100.0 3rd Qu.:6.00 3rd Qu.:4.00 3rd Qu.:0.00
## Max. :100.0 Max. :7.00 Max. :7.00 Max. :1.00
## term estimate std.error statistic p.value
## 1 (Intercept) 93.4 1.572 59.4 0.00e+00
## 2 RConserv -4.1 0.368 -11.2 9.48e-28
## 3 GOP -26.5 1.587 -16.7 2.82e-57
## r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
## 1 0.325 0.324 23.1 336 9.28e-120 3 -6365 12738 12759
## deviance df.residual
## 1 741815 1394
\[ \begin{align} \text{Obama thermometer} = \beta_0 &+ \beta_1 (\text{Respondent conservatism}) \\ & + \beta_2 (\text{GOP respondent})\\ & + \beta_3 (\text{Respondent conservatism}) (\text{GOP respondent}) \\ & + e \end{align} \]
## term estimate std.error statistic p.value
## 1 (Intercept) 92.25 1.640 56.24 0.00e+00
## 2 RConserv -3.81 0.388 -9.81 5.26e-22
## 3 GOP -11.07 6.684 -1.66 9.79e-02
## 4 RConserv:GOP -2.86 1.201 -2.38 1.75e-02
## r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
## 1 0.328 0.326 23 226 1.15e-119 4 -6362 12735 12761
## deviance df.residual
## 1 738815 1393
GOP = 0
\[ \begin{align} E(\text{Obama thermometer}) = 92.255 & -3.805 (\text{Respondent conservatism}) -11.069 (0)\\ & -2.856 (\text{Respondent conservatism} \times 0) \\ = 92.255 & -3.805 (\text{Respondent conservatism}) \end{align} \]
GOP = 1
\[ \begin{align} E(\text{Obama thermometer}) & = (92.255 -11.069 (1)) + (-3.805 -2.856 (\text{Respondent conservatism} \times 1)) \\ & = 81.186 -6.661 (\text{Respondent conservatism}) \end{align} \]
## term estimate std.error statistic p.value
## 1 (Intercept) 92.25 1.598 57.7 0.00e+00
## 2 RConserv -3.81 0.378 -10.1 7.87e-23
## term estimate std.error statistic p.value
## 1 (Intercept) 81.19 6.98 11.62 1.90e-26
## 2 RConserv -6.66 1.22 -5.44 1.04e-07
\[ \begin{align} \text{Obama thermometer} = \beta_0 &+ (\beta_1 + \beta_3 \text{GOP}) (\text{Respondent conservatism}) \\ & + \beta_2 (\text{GOP respondent}) + e \\ = &\beta_0 + \psi_1 (\text{Respondent conservatism}) + \beta_2 (\text{GOP respondent}) + e \end{align} \]
Point estimate
## [1] -13.9Standard error
\[\hat{\sigma}_{\hat{\psi}_1} = \sqrt{\widehat{\text{Var}(\hat{\beta}_1)} + (\text{GOP})^2 \widehat{\text{Var}(\hat{\beta_3})} + 2 (\text{GOP}) \widehat{\text{Cov}(\hat{\beta}_1 \hat{\beta}_3)}}\]
## (Intercept) RConserv GOP RConserv:GOP
## (Intercept) 2.691 -0.574 -2.691 0.574
## RConserv -0.574 0.151 0.574 -0.151
## GOP -2.691 0.574 44.677 -7.797
## RConserv:GOP 0.574 -0.151 -7.797 1.442
## [1] 1.14linearHypothesis(obama_ideo_gop, "RConserv + RConserv:GOP")## Linear hypothesis test
##
## Hypothesis:
## RConserv + RConserv:GOP = 0
##
## Model 1: restricted model
## Model 2: ObamaTherm ~ RConserv * GOP
##
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 1394 757039
## 2 1393 738815 1 18225 34.4 5.7e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
linearHypothesis(obama_ideo_gop, "GOP + 7 * RConserv:GOP")## Linear hypothesis test
##
## Hypothesis:
## GOP + 7 RConserv:GOP = 0
##
## Model 1: restricted model
## Model 2: ObamaTherm ~ RConserv * GOP
##
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 1394 821850
## 2 1393 738815 1 83036 157 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
\[ \begin{align} \text{Obama thermometer} = \beta_0 &+ \beta_1 (\text{Respondent conservatism}) \\ & + \beta_2 (\text{Obama conservatism})\\ & + \beta_3 (\text{Respondent conservatism}) (\text{Obama conservatism}) \\ & + e \end{align} \]
## term estimate std.error statistic p.value
## 1 (Intercept) 117.12 2.972 39.41 8.00e-229
## 2 RConserv -14.94 0.600 -24.88 2.52e-113
## 3 ObamaConserv -6.73 0.929 -7.25 7.06e-13
## 4 RConserv:ObamaConserv 2.81 0.182 15.40 1.53e-49
## r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
## 1 0.451 0.45 20.8 381 8.3e-181 4 -6221 12452 12478
## deviance df.residual
## 1 603467 1393
\[p(\text{Survival}) = \frac{e^{\beta_0 + \beta_{1}\text{Age} + \beta_{2}\text{Sex}}}{1 + e^{\beta_0 + \beta_{1}\text{Age} + \beta_{2}\text{Sex}}}\]
## term estimate std.error statistic p.value
## 1 (Intercept) 1.27727 0.23017 5.55 2.87e-08
## 2 Age -0.00543 0.00631 -0.86 3.90e-01
## 3 Male -2.46592 0.18538 -13.30 2.26e-40
## null.deviance df.null logLik AIC BIC deviance df.residual
## 1 965 713 -375 756 770 750 711
\[p(\text{Survival}) = \frac{e^{\beta_0 + \beta_{1}\text{Age} + \beta_{2}\text{Sex} + \beta_3 (\text{Age} \times \text{Sex})}}{1 + e^{\beta_0 + \beta_{1}\text{Age} + \beta_{2}\text{Sex} + \beta_3 (\text{Age} \times \text{Sex})}}\]
## term estimate std.error statistic p.value
## 1 (Intercept) 0.5938 0.3103 1.91 0.05569
## 2 Age 0.0197 0.0106 1.86 0.06240
## 3 Male -1.3178 0.4084 -3.23 0.00125
## 4 Age:Male -0.0411 0.0136 -3.03 0.00241
## null.deviance df.null logLik AIC BIC deviance df.residual
## 1 965 713 -370 748 767 740 710
\[ \begin{align} \text{Obama thermometer} = \beta_0 &+ f_1 (\text{Respondent conservatism}) \\ & + f_2 (\text{Obama conservatism}) + e \end{align} \]
\[ \begin{align} \text{Obama thermometer} = \beta_0 &+ f_1 (\text{Respondent conservatism}) \\ & + f_2 (\text{Obama conservatism})\\ & + f_3 (\text{Respondent conservatism} \times \text{Obama conservatism}) + e \end{align} \]
##
## Family: gaussian
## Link function: identity
##
## Formula:
## ObamaTherm ~ s(RConserv, ObamaConserv, k = 5)
##
## Parametric coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 69.628 0.555 125 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## s(RConserv,ObamaConserv) 3.98 4 290 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.453 Deviance explained = 45.5%
## GCV = 432.19 Scale est. = 430.65 n = 1397